Basic Percentages (AQA GCSE Maths)

What is a percentage?

• “Per-cent” simply means “ ÷ 100” (or “out of 100”)
• You can think of a percentage as a standardised way of expressing a fraction – by always expressing it “out of 100”
• That means it is a useful way of comparing fractions e.g.

½ = = 50% 

⅖ =  = 40%

¾ =  = 75%

How do I work out basic percentages of amounts?

• You can use simple equivalences to calculate percentages of amounts without a calculator
  • 50% =  so you can divide by 2
  • 25% = so you can divide by 4
  • 20% =  so you can divide by 5
  • 10% =  so you can divide by 10
  • 5% = so you can find 10% then divide by 2
  • 1% =  so you can divide by 100 etc.

• You can then build up more complicated percentages such as 17% = 10% + 5% + 2 x 1%

How do I find any percentage of an amount?

• Method 1: You can find any percentage of an amount by dividing by 100 and multiplying by the given %
  • 23% of 40 is 40 ÷ 100 = 0.4, multiply by 23:  0.4 × 23 = 9.2
• Method 2: To find “a percentage of X”: multiply X by the "decimal equivalent" of that percentage (percentage ÷ 100)
  • for example,  23% of 40 is 40 x 0.23 = 9.2
• To find “A as a percentage of B”: do A ÷ B to get a decimal, then x 100, e.g.
  • for example, to find 26 as a percentage of 40 first do 26 ÷ 40 = 0.65, then x 100 to get 65%
    • 26 is 65% of 40

How do I increase or decrease by a percentage?

• Identify “before” & “after” quantities
• If working in percentages, add to (or subtract from) 100 
  • a percentage increase of 25% is 100 + 25 = 125% of the "before" price
    • Add 25% to the original amount
  • a percentage decrease of 25% is 100 - 25 = 75% of the "before" price
    • Subtract 25% from the original amount
• A multiplier, p, is the decimal equivalent of a percentage increase or decrease
  
  • The multiplier for a percentage increase of 25% is p = 1 + 0.25 = 1.25
    • Multiply the original amount by the multiplier, 1.25, to find the new amount
  • The multiplier for a percentage decrease of 25% is p = 1 - 0.25 = 0.75
    • Multiply the original amount by the multiplier, 0.75, to find the new amount
• The amount "before" and the amount "after" a percentage change are related by the formula "before" × p = "after"
  • where p is the multiplier

How do I find a percentage change?

• Method 1: rearrange the formula "before" × p = "after" to make p (the multiplier) the subject
  • p =  
  • Calculate p and interpret its value
    • p = 1.02 shows a percentage increase of 2%
    • p = 0.97 shows a percentage decrease of 3%
• Method 2: Use the formula that the "percentage change" is  
  • A positive value is a percentage increase 
  • A negative value is a percentage decrease
• The same formula can be used for percentage profit (or loss)
  • the "cost" price is the price a shop has to pay to buy something and the "selling" price is how much the shop sells it for

• You can identify whether there is a profit or loss 
  • cost price < selling price = profit (formula gives a positive value)
  • cost price > selling price = loss (formula gives a negative value)